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Choice of a Short-term Prediction Model for Patient Discharge Before Noon

A Walk-Through of ARIMA Model

Berrios-Montero, Rolando A., PhD

doi: 10.1097/HCM.0000000000000262

Hospital leaders encourage morning discharge of patients to boost patient flow. This work presents a detailed process of a building model for forecasting patient discharge before noon applying the Box-Jenkins methodology using weekly historic data. Accurately forecasting is of crucial importance to plan early discharge activities, influenced by the fluctuations in daily discharges process. The objective is to find an appropriate autoregressive integrated moving average (ARIMA) model for forecasting the rate of patients out by noon based on the lowest error in a statistical forecast by applying the mean absolute percentage error. The results obtained demonstrate that a nonseasonal ARIMA model classified as ARIMA(2,1,1) offers a good fit to actual discharge-before-noon data and proposes hospital leaders short-term prediction that could facilitate decision-making process, which is important in an uncertain health care system environment.

Author Affiliation: Health System Strengthening, Chemonics International, Inc, Arlington, Virginia.

The author has no funding or conflicts of interest to disclose.

Correspondence: Rolando A. Berrios-Montero, PhD, Health System Strengthening, Chemonics International, Inc, 251 18th St South, Arlington, VA 22202 (

R.A.B.-M. is a performance and process improvement director for USAID Global Health Supply Chain Program, which involves redesigning many of process systems and implementing performance measures to ensure patients have access to the lifesaving medications they need.

PATIENT DISCHARGE HAS been identified as one of key processes to reduce the inpatient length-of-stay metric. Since the introduction of the Affordable Care Act in March 2010, US hospitals have been incentivized to discharge patients as soon as possible to be, but still increase focus on high-value care. A rationale for this initiative is to facilitate patient throughput. In other words, the earlier in the day a patient is discharged, the faster a patient who was admitted through the emergency department can be assigned an inpatient bed. Therefore, treatment can begin sooner, reducing the risk of patients getting worse outcomes and increasing patient satisfaction while at the hospital.

Patient discharge before noon (DBN) is a multidisciplinary improvement initiative that may create financial pressure among caregivers and could lead to worse patient outcomes. The DBN intervention should consider a standardized discharge system including scheduled interdisciplinary rounds, electronic communication tool,1 patient identification for discharge, communication with patient's relatives, and coordination with pharmacy, transportation, and patient facility destination. Nevertheless, the principal factor considered when planning for health care capacity admission is the rate at which patients are discharged.1 US hospital leaders realize that in order to successfully discharge patients in the morning rather than late in the day they need to leverage historic performance data and data analytics to identify DBN barriers and areas of opportunity for improvement.

There have been previous researches that include optimizing critical discharge processes, such as multidisciplinary rounding, patient care transition education, placing prescriptions to outpatient pharmacies, and patient destination transportation.1-4 Whereas there has been significant research on improving patient discharge, there still is a literature gap. Administrators and hospital leaders still do not know how to best use historical data to forecast DBN to support decision making when redesigning patient discharge processes or implementing new activities to improve discharge performance.5 Hospital leaders must emphasize on the development of a discharge planning system including how hospitalist will write or issue a discharge orders (not conditional discharge orders) so patients can be identified on the DBN forecast. Appropriate prediction of patients DBN plays an important role in reducing unnecessary discharge processes on patients who are not ready to be discharged, resulting in caregivers not devoting their attention to patients who need it most.

This study analyzes discharge performance time-series data by the application of the autoregressive integrated moving average (ARIMA) model developed by George Box and Gwilym Jenkins6 in 1976. The methodology concentrates on a systematic approach of a set of procedures for identifying, fitting, checking, and forecasting using ARIMA time-series models. This process is also referred to as the Box-Jenkins method.

The objective of this work is to predict future behavior of a time series on DBN. The time-series model is a solid choice for discharge performance forecasting in terms of multiple discharge data attributes such as randomness, cyclic behavior, or trend.7 The ability of hospital leaders to predict discharge performance is significant to decision-making process aimed at producing improvement, providing quality patient care, and meeting the expected performance goals. Moreover, applying this methodology can support hospital leaders and caregivers to initiate a conversation for identifying key discharge drivers to improve process improvement initiatives. For instance, Figure 1 shows an individual and moving range (I-MR) chart to examine DBN process behaviors, where week 36 shows a process that is out of control and a trend of DBN slightly upward with no evident seasonality component. It is a cyclical time series because the original data trended graph displays up-and-down movement repeated over 52 weeks, and each week did not show the same pattern. The I-MR chart for DBN shows its DBN performance trend where there may be some issues that need to be addressed. This illustration identifies and corrects the factors contributing to variation.

Figure 1

Figure 1

The purpose of this work is to analyze the performance of ARIMA models, describe its construction, discuss results, and select the best forecast of DBN ratio (based on measuring forecasting accuracy such as mean absolute percentage error [MAPE]). While the application of various ARIMA models has not previously been reviewed, this study helps hospital leaders and practitioners by reviewing this useful prediction tool to identify interventions that may affect the DBN trend or seasonality.

This study will be limited to the application of the Box-Jenkins methodology where effects of the trend of DBN performance over time can be observed. The data set is gathered from electronic time stamps of patient's medical records for 17 clinical units at a 900-bed health care hospital system located in northern Virginia, United States, serving an urban population. Patient discharge data are collected for 52 weeks, from July 2016 to June 2017, including 34 185patients where 4965 patients were successfully DBN.

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The ARIMA model is a popular time-series forecasting technique. Time-series models explain the values in a future series based on its own historical data set. Its application works best in short-term forecasting when data exhibit a stable pattern over time.8 The time series simulated here will be used to predict noon discharge patients.

The ARIMA model describes the behavior in a stationary time series as a function of autoregressive (AR) and moving average (MA) parameters. It explicitly includes differencing in the model formulation. Specifically, the 3 types of model parameters are the AR parameters (p), the number of time the series is differenced (d), and MA parameters (q).6

Discharge performance time series can be stationary or nonstationary. Its mean and SD are constant over time. This DBN time series is stochastic, and if stationary, it is possible to use the ARIMA(p,d,q) without excluding the possibility of using mixed autoregressive moving average (ARMA(p,q)).

Autocorrelation. Autocorrelation is a numerical representation of the similarities between original data points and a time lagged of itself; also, it is used to detect data nonrandomness. The correlation is between 2 data points of the same variable at times ti and ti + k, where k = 1, 2,…. Autocorrelation is a correlation coefficient between adjacent error terms. The autocorrelation parameter is introduced as

Because autocorrelation ranges from +1 to −1, a value close to +1 indicates positive correlation, and a value close to −1 implies negative correlation.

AR models. Autoregressive describes a process in which the observations at a given moment are predictable from the previous process observations plus an error term. An AR(p) is expressed as follows:

where Yt = time series under consideration

βP = the AR parameter of order p

Yt-P = the time series lagged p period

εt = model error

t = time period

AR(1) represents any observed value; Yt can be explained by a function of its previous period Yt-1 plus random error εt. For example, if the estimated value of AR(1) is 0.20, then the number preceding is 1, and the observed value would be related to 20% of its previous value.

MA models. Moving average parameters relate what happens in period t only to the random errors that occurred in past time periods rather than to Yt-1, Yt-2, Yt-3 as in the AR approaches. The following expression may be used to develop an MA model with q term: MA(q).

where Yt = time series under consideration

µ = constant process mean

γq = coefficient to be estimated

εt-q = the errors in previous model time periods

t = time period

Mixed models. The AR with parameter p is a linear combination of preceding number Y values, whereas the MA with parameter q is the linear combination where errors between consecutive data points are accounted. ARMA(p,q) could be expressed as follows:

The series could be made stationary by logarithmic transformation or taking first difference. Taking appropriate differences, an ARIMA model can be used once stationarity is achieved. Integration refers to the rate of change process to produce the forecast. An ARIMA model is usually stated as ARIMA(p,d,q).

It is proposed to find a forecast model to predict future weekly patient discharge performance using the ARIMA. Therefore, it is necessary to follow 4 steps to build an ARIMA model8: (1) model identification, (2) estimation, (3) diagnostic checking, and (4) forecasting.

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With the assistance of Minitab Statistical Software (Minitab Inc, State College, Pennsylvania), the ARIMA model was built following the previous steps explained in the Methodology section.

Model identification.Determine if the time series is stationary by developing and considering the autocorrelation function (ACF) and partial autocorrelation function (PACF) graphs. The outcome of these graphs will support practitioners in determining whether differencing is necessary to stationarize the series. Determine ARIMA model to forecast.

The dynamic dependence of Yt on its past values is determined by defining the backshift operator, which causes the observation that it multiplies to be backward in time by 1 period.

For the discharge performance time series Yt, B is the backward shift operator, n equals any integer, and t is any period. For example, the first difference of Y can be expressed by y for any t.

Both ACF and PACF in Figure 2 show that the discharge performance time series needs to be made stationary. The original data are not stationary because they show an upward trend. The ACF function values 0.4600, 0.4653, and 0.4505 show significant peaks in lags 1, 2, and 4, respectively, with subsequent values (0.4007, …, 0.0688) that did not decay rapidly, which is a typical AR process. Moreover, the PACF generates confidence bands at 5% for the hypothesis that the correlations are equal to zero. The vertical lines that overpass the 5% autocorrelation significance limits indicate correlation between points lagged by 1 and 4 periods in time.

Figure 2

Figure 2

The time-series data need to be made stationary because neither upward nor downward trends should exist. This is a simple process where 1 or more series differences are taken to eliminate the trend.

where the new series Wt is obtained by subtracting from each observation of zt in the previous observation. Figure 3 displays the ACF and PACF analysis outcome on the differenced data. Both functions have high peaks for lag 1, which suggests AR(1). Partial autocorrelation function eventually falls to zero, indicating that the processes of moving and AR averages are occurring. Moreover, the PACF spikes up to lag 1, and lag 3 overpassing the 95% confidence intervals suggests an MA(1) or MA(2). ARIMA modeling can be a bit iterative.

Figure 3

Figure 3

Estimation.Estimate the parameters for a tentative model by determining and choosing the p,d,q order based on ACF and PACF.

Considering the ARIMA(1,1,1) model for the discharge performance time series by omitting the constant term, the following equations for the model are developed:

Applying the first difference of Y, then

Applying the backshift operator B, the equation can be rewritten, which defines a linear regression model with ARIMA(1,1,1).

Table 1 shows additional outcomes for orders p,d,q of ARIMA models where ARIMA(2,1,1) is the most suitable for our time series.

Table 1

Table 1

Diagnostic checking.The model adequacy test is checked by considering whether the properties of the ARIMA model residuals are normally distributed. The model check is conducted by applying Ljung-Box Q-statistic where the P value is less than α (α = .05). If P value is not less than .05, then the new model is specified and tested again for adequacy (P < α). The process is repeated until an adequate model meets the P value criteria.

In Figure 4, residuals do not appear to be correlated because none of the data displayed in the vertical lines are within statistical significance bounds.

Figure 4

Figure 4

Table 2 shows the final estimate parameters output from Minitab. ARIMA(2,1,1) are all significant at the 5% level, indicating that this is a suitable forecasting model. The sum of squares (SE Coef) between each original data and its estimated value ranges from 0.146 and 0.297, which are quite small. Also, notice the Ljung-Box χ2 statistic P values are well above .05, which indicates the desirable result of nonsignificance meaning that the residuals are independent. The Ljung-Box χ2 statistics show no correlations between points with a difference of 12, 24, 36, or 48 lags.

Table 2

Table 2

Forecasting.Calculate future values [ZERO WIDTH SPACE][ZERO WIDTH SPACE] using the established final model in the previous steps.

ARIMA(2,1,1) model seems to be a good fit for the discharge data simulated to forecast future performance values. Figure 5 shows the forecast profile for the next 12 weeks, which could be used to predict expected discharge performance values ranging from 0.16 (week 53) to 0.15 (week 60). The ARIMA model provides forecasts with confidence bands of 95% with the predicted values [ZERO WIDTH SPACE][ZERO WIDTH SPACE] slightly lower than the previous 20 weeks.

Figure 5

Figure 5

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Measure of Prediction Accuracy

The ARIMA model that has the minimum deviation is considered as the best forecasting model.9 Accuracy of forecasting models is decisive when managing short-term prediction while planning DBN processes that may have an impact on hospital operation results. Therefore, this DBN forecasting study has used MAPE to examine the performance of ARIMA models. Mean absolute percentage error is given by the following equation where it takes the absolute DBN error and divided by the actual DBN value gathered to obtain a forecast percent error:

Xt equals actual value, xt equals the projected value, and n is the total number of observations considered in the model. The lowest MAPE results computed are 7.1% for ARIMA(2,1,1) followed by 7.6% and 8.5% for ARIMA(2,0,2) and ARIMA(1,1,1), respectively.

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A practical ARIMA model for forecasting the short-term performance of discharging patients from a hospital facility by noon, of the order (2,1,1), is found to be the best to fit the time series of discharge performance up to 8 weeks. Also, it demonstrated that the forecasting model selection is superior to MAPE. It is confirmed that the estimated model parameters are significantly different from zero. The diagnostic is applied to ARIMA(2,1,1) residuals, and the results indicate that the residuals are independent and normally distributed. Also, the predicted data show reliable agreement with the actual data. There are ongoing initiatives throughout hospitals to improve patient throughput, and the results obtained here show that to understand why goals of BDN are not met hospital leaders must understand historical data, barriers, and the activities associated to such performance.

Forecast accuracy depends on daily workflow and frontline staff experience. Early discharge patient identification triggers discharge steps such as morning huddle and patient-nurse-physician rounds. However, there are many barriers faced during the process of getting patients out by noon, such as

  • no available bed placement at skilled nursing facilities;
  • waiting on attending physicians to see patient;
  • discharge order is conditional;
  • waiting for prescriptions to be filled;
  • additional tests are ordered;
  • waiting for insurance authorization;
  • patient condition status changes after discharge order is written;
  • patient does not want to leave;
  • family cannot take patient until late in the day;
  • family not prepared to take care of patient at home; and
  • waiting for home care equipment.

These barriers are considerations for discharge process and to facilitate timely discharge. Therefore, this information could be used by hospital leaders to reflect how discharge processes have been executed and what significant corrective actions should be implemented in the short term to improve performance and achieve desired goals.

Whereas hospital leaders encourage DBN to improve patient throughput, it is worthwhile mentioning that forecasting analyses could help administrators make decisions regarding patient DBN, improving overall patient flow and reducing the patient's length of stay.

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ARIMA; Box-Jenkins methodology; forecasting; patient discharge by noon; time series

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